1. ANGULAR MOMENTUM TRANSPORT BY MAGNETOHYDRODYNAMIC TURBULENCE Gordon Ogilvie University of Cambridge TACHOCLINE DYNAMICS 11.11.04

2. INTRODUCTION SOME TACHOCLINE ISSUES (Tobias 2004) ► sources of instability : HD and MHD ► nonlinear development ► turbulence and turbulent transport : HD and MHD SOME ACCRETION DISC ISSUES ► differential rotation and AM transport ► HD and MHD instabilities ► turbulence and turbulent transport : HD and MHD

3. COMPARISON TACHOCLINE ACCRETION DISC ► thin ► thin ► differentially rotating ► differentially rotating

4. COMPARISON TACHOCLINE ACCRETION DISC ► thin ► thin ► differentially rotating ► differentially rotating ► magnetized (probably) ► magnetized (probably) ► turbulent (probably) ► turbulent (probably) ► large-scale dynamo? ► large-scale dynamo?

5. COMPARISON TACHOCLINE ACCRETION DISC ► thin ► thin ► differentially rotating ► differentially rotating ► magnetized (probably) ► magnetized (probably) ► turbulent (probably) ► turbulent (probably) ► large-scale dynamo? ► large-scale dynamo? ► highly subsonic ► highly supersonic ► strong stable stratification? ► weak or no stratification?

6. COMPARISON TACHOCLINE ACCRETION DISC ► thin ► thin ► differentially rotating ► differentially rotating ► magnetized (probably) ► magnetized (probably) ► turbulent (probably) ► turbulent (probably) ► large-scale dynamo? ► large-scale dynamo? ► highly subsonic ► highly supersonic ► strong stable stratification? ► weak or no stratification? ► difficult to resolve ► difficult to resolve ► difficult to simulate ► difficult to simulate

7. ANGULAR MOMENTUM TRANSPORT GENERAL ► anisotropic motion (Reynolds stress) ► anisotropic magnetic fields (Maxwell stress) ► non-axisymmetric gravitational fields LARGE-SCALE STRUCTURES SMALL-SCALE FEATURES ► spiral arms / shocks ► waves ► vortices ► turbulence

8. SHEARING SHEET ► local model of a differentially rotating disc ► uniform rotation Ωez plus uniform shear flow –2Axey ► appropriate for studies of thin discs

9. MAGNETOROTATIONAL INSTABILITY OPTIMAL MODE (‘channel flow’) ► layer analysis (incompressible ideal fluid, ρ= μ0 = 1) u b ► exact nonlinear solution but unstable (Goodman & Xu 1994)

10. MAGNETOROTATIONAL INSTABILITY NONLINEAR DEVELOPMENT (A. Brandenburg)

11. MAGNETOROTATIONAL INSTABILITY NONLINEAR DEVELOPMENT

12. MAGNETOROTATIONAL INSTABILITY NONLINEAR DEVELOPMENT

13. ENERGY AND ANGULAR MOMENTUM ENERGY EQUATION (shearing sheet) ► in either growing instability or saturated turbulence, ► AM transport down the gradient of angular velocity ► very natural outcome of MHD instabilities ► contrast (e.g.) convective instability or forced turbulence

14. TURBULENCE MODELS EDDY-VISCOSITY MODEL (von Weizsäcker 1948) VISCOELASTIC MODEL (O 2001; O & Proctor 2003) REYNOLDS-MAXWELL STRESS MODELS (Kato; O 2003)

15. SOME CONTROVERSIES ► ‘viscosity’ ► ‘alpha viscosity’ ► AM transport by convection ► nonlinear hydrodynamic shear instability ► baroclinic / Rossby-wave instability

16. CONTINUOUS SPECTRUM INTRODUCTION ► cf. Friedlander & Vishik (1995); Terquem & Papaloizou (1996) ► problems with a normal-mode approach in shearing media ●modes may require confining boundaries ●entirely absent (ky≠0) in the shearing sheet ●do not describe parallel shear flow instability ► continuous spectrum and non-modal localized approaches ●derive sufficient conditions for instability ●contain many of the most important instabilities

17. CONTINUOUS SPECTRUM LINEAR THEORY IN IDEAL MHD ► arbitrary reference state ► Lagrangian displacement ξ

18. CONTINUOUS SPECTRUM BASIC STATE ► steady and axisymmetric ► cylindrical polar coordinates (s,φ,z) ► differential rotation ► toroidal magnetic field SOLUTIONS

19. CONTINUOUS SPECTRUM ASYMPTOTIC LOCALIZED SOLUTIONS ► envelope localized near a point (s0,z0) ► plane-wave form with many wavefronts ► finite frequency and vanishing group velocity ► ‘frozen wavepacket’

20. CONTINUOUS SPECTRUM REQUIRED ORDERING

21. CONTINUOUS SPECTRUM LOCAL DISPERSION RELATION

22. CONTINUOUS SPECTRUM CASE OF ZERO MAGNETIC FIELD ► Høiland (1941) stability criteria ► necessary and sufficient for axisymmetric disturbances

23. CONTINUOUS SPECTRUM LIMIT OF WEAK MAGNETIC FIELD ► Papaloizou & Szuszkiewicz (1992) stability criteria ► necessary but not sufficient for stability

24. CONTINUOUS SPECTRUM CASE OF ZERO ANGULAR VELOCITY ► Tayler (1973) stability criteria ► necessary and sufficient

25. APPLICATION TO ACCRETION DISCS ► appropriate ordering scheme for a thin disc reveals ● MRI (unavoidable) ●magnetic buoyancy instability (possible) ► allows an understanding of the nonlinear state? differential rotation MRI

26. APPLICATION TO THE TACHOCLINE ► appropriate ordering schemes are unclear (to me) ► assume overwhelming stable stratification

27. APPLICATION TO THE TACHOCLINE ► appropriate ordering schemes are unclear (to me) ► assume overwhelming stable stratification ● weak B: MRI when (NB: no MRI in 2D) ●Ω=0 : Tayler (m=1) when ● suppressed at the poles if ● cf. Cally (2003) (but not requiring mode confinement) ► conclusions change under weaker stratification ● sensitivity to radial gradients; magnetic buoyancy

28. REMARKS ADVANTAGES ► algebraic character of eigenvalues and eigenvectors ► strictly local character, independent of BCs ► deals easily with complicated 2D basic states PROPER JUSTIFICATION ► prove existence of continuous spectrum ► asymptotic treatment of non-modal disturbances ► justifies ‘local analysis’ for a restricted class of disturbances

29. REMARKS NOTES OF CAUTION ► misses truly global instabilities ► neglects the role of turbulent stresses in the basic state ► neglects diffusion (double / triple) in the perturbations ●Acheson (1978); Spruit (1999); Menou et al. (2004)

30. SUMMARY ► analogies are imperfect but of some value ► angular momentum transport and energy arguments ► differences between HD and MHD systems ► MRI optimized for AM transport down the gradient of angular velocity but of limited applicability in the Sun ► methods for analysing linear instabilities ► continuous spectrum contains many of the important ones ► methods for understanding and modelling turbulent states